Let \(u(x)\colon \mathbf{R}^d \rightarrow \mathbf{R}^d\) such that \(\sup_x |u(x)| \leq K \). Define \(X_t\) by

\[dX_t = u(X_t)dt + \sigma dB_t\]

for \(\sigma >0\) and \(X_0=x\).

For any open set \( A \subset \mathbf{R}^d\) assume that you know that \(P(B_t \in A) >0)\) show that the same holds for \(X_t\).

Hint: Start by showing that \(\mathbf{E}[ f(x+ \sigma B_t)] = \mathbf{E}[\Lambda_t f(X_t)]\) for some process \(\Lambda_t\) and any function \(f\colon \mathbf{R}^d\rightarrow \mathbf{R}\). Next show that \(\mathbf{E}[\Lambda_t^2] < \infty\)